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Polynomials - Relationship between Zeroes and Coefficients of a Polynomial

Grade 10CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A zero of a polynomial p(x)p(x) is a value kk such that p(k)=0p(k) = 0. Geometrically, the zeroes of a polynomial are the xx-coordinates of the points where the graph of y=p(x)y = p(x) intersects the xx-axis.

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For a linear polynomial ax+bax + b, the zero is simply −ba-\frac{b}{a}, which represents the point where the straight line graph crosses the xx-axis.

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A quadratic polynomial ax2+bx+cax^2 + bx + c can have at most 2 zeroes. Its graph is a parabola that opens upwards if a>0a > 0 or downwards if a<0a < 0. The relationship between the zeroes (α,β\alpha, \beta) and coefficients (a,b,ca, b, c) is fundamental to solving algebraic equations.

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For a quadratic polynomial ax2+bx+cax^2 + bx + c, the sum of zeroes is equal to −coefficient of xcoefficient of x2-\frac{\text{coefficient of } x}{\text{coefficient of } x^2} and the product of zeroes is equal to constant termcoefficient of x2\frac{\text{constant term}}{\text{coefficient of } x^2}.

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A cubic polynomial ax3+bx2+cx+dax^3 + bx^2 + cx + d can have at most 3 zeroes. Visually, its graph can cross the xx-axis at up to three distinct points, reflecting the three possible roots.

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For a cubic polynomial, the relationships are: the sum of zeroes is −ba-\frac{b}{a}, the sum of the products of zeroes taken two at a time is ca\frac{c}{a}, and the product of zeroes is −da-\frac{d}{a}.

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To form a quadratic polynomial when the sum of zeroes (SS) and product of zeroes (PP) are known, we use the structure k[x2−Sx+P]k[x^2 - Sx + P], where kk is a non-zero real constant.

📐Formulae

Linear Zero: x=−bax = -\frac{b}{a}

Quadratic Sum of Zeroes: α+β=−ba\alpha + \beta = -\frac{b}{a}

Quadratic Product of Zeroes: αβ=ca\alpha\beta = \frac{c}{a}

Quadratic Polynomial Formation: p(x)=k[x2−(α+β)x+αβ]p(x) = k[x^2 - (\alpha + \beta)x + \alpha\beta]

Cubic Sum of Zeroes: α+β+γ=−ba\alpha + \beta + \gamma = -\frac{b}{a}

Cubic Sum of Product in pairs: αβ+βγ+γα=ca\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}

Cubic Product of Zeroes: αβγ=−da\alpha\beta\gamma = -\frac{d}{a}

💡Examples

Problem 1:

Find the zeroes of the quadratic polynomial x2+7x+10x^2 + 7x + 10, and verify the relationship between the zeroes and the coefficients.

Solution:

Step 1: Factorize the polynomial by splitting the middle term: x2+5x+2x+10=x(x+5)+2(x+5)=(x+2)(x+5)x^2 + 5x + 2x + 10 = x(x + 5) + 2(x + 5) = (x + 2)(x + 5) Step 2: Set the factors to zero to find the zeroes: x+2=0⇒x=−2x + 2 = 0 \Rightarrow x = -2 and x+5=0⇒x=−5x + 5 = 0 \Rightarrow x = -5. So, α=−2\alpha = -2 and β=−5\beta = -5. Step 3: Verify sum of zeroes: α+β=−2+(−5)=−7\alpha + \beta = -2 + (-5) = -7. From formula, −ba=−71=−7-\frac{b}{a} = -\frac{7}{1} = -7. Step 4: Verify product of zeroes: αβ=(−2)(−5)=10\alpha\beta = (-2)(-5) = 10. From formula, ca=101=10\frac{c}{a} = \frac{10}{1} = 10.

Explanation:

We first use factorization to find the actual roots (zeroes). Then, we calculate the sum and product of these roots and compare them with the values derived from the coefficients using the formulas −ba-\frac{b}{a} and ca\frac{c}{a} to confirm they match.

Problem 2:

Find a quadratic polynomial, the sum and product of whose zeroes are 14\frac{1}{4} and −1-1, respectively.

Solution:

Step 1: Identify the given values: Sum of zeroes S=α+β=14S = \alpha + \beta = \frac{1}{4}. Product of zeroes P=αβ=−1P = \alpha\beta = -1. Step 2: Use the polynomial formation formula: p(x)=k[x2−Sx+P]p(x) = k[x^2 - Sx + P]. Step 3: Substitute the values: p(x)=k[x2−14x+(−1)]=k[x2−14x−1]p(x) = k[x^2 - \frac{1}{4}x + (-1)] = k[x^2 - \frac{1}{4}x - 1]. Step 4: To remove the fraction, let k=4k = 4: p(x)=4(x2−14x−1)=4x2−x−4p(x) = 4(x^2 - \frac{1}{4}x - 1) = 4x^2 - x - 4.

Explanation:

To construct a polynomial from its zeroes, we use the standard form x2−(sum)x+(product)x^2 - (\text{sum})x + (\text{product}). If there are fractions, we multiply the entire polynomial by a constant kk (usually the denominator) to obtain an expression with integer coefficients.